for given \(\tau \in [0,T]\) , \(\varepsilon >0\)  and admissible control \(u(\cdot )\) . (REF ) can be regarded as the stochastic version corresponding to the error estimate (1.8) in [1]}, where the definition of two-parameter family of admissible controls \(u_{\tau \varepsilon }(\cdot )\)  follow the idea to find the set \(E_{\tau \varepsilon }\)  with small measure on which \(u_{\tau \varepsilon }(\cdot )\)  minimize the \(\mathcal {H}\) -function and keep identical with \(u(\cdot )\)  outside \(E_{\tau \varepsilon }\) . In (REF ) the term \(\mathbb {E}\left[ \int _{E_{\tau \varepsilon }}\Delta _{u}\mathcal {H}(t)dt\right] \le 0\)  that characterizes the extent to which \(u(\cdot )\)  deviates from satisfying the necessary condition for optimality, and the equality holds if and only if \(u(\cdot )\)  exactly satisfies the stochastic maximum principle.
As soon as such deviation is strictly less than 0, we can further understand it to be order of \(\varepsilon \)  small enough by Lebesgue's differentiation theorem. Then, for sufficiently small \(\varepsilon >0\) , it is not difficult to observe from (REF ) that \(u_{\tau \varepsilon }(\cdot )\)  can reduce \(J\)  by the extent of order \(O((1-\varepsilon ^{\frac{1}{2}})\varepsilon )\) . However, the decreasing of \(J\)  in the deterministic case is more sharp by the extent of order \(O((1-\varepsilon )\varepsilon )\) , since the remainder term on the right-hand side of (1.8) in [1]} is of order \(O(\varepsilon ^{2})\)  but its counterpart, as we can see in (REF ), is of order \(O(\varepsilon ^{\frac{3}{2}})\) .
What should be emphasized is that, since (REF ) is a mechanism to reduce \(J\)  in each iteration, the multiplier \(C\)  in the remainder term of (REF ) must be uniform across all admissible controls \(u(\cdot )\)  and independent of the choices of \(\tau \) . So in the rigorous proof of (REF ) we essentially show that \(C\)  only depends on the dimensions of \(X\) , \(W\)  and prescribed parameters imposed on the coefficients and control domains. To this end, it needs mild assumptions that the second-order partial derivatives of all the coefficients in the concerned systems are Lipschitz in \(x\)  uniformly in \((t,u)\) .
